Mixed Convolved Action Variational Methods for Poroelasticity

Although Lagrangian and Hamiltonian analytical mechanics represent perhaps the most remarkable expressions of the dynamics of a mechanical system, these approaches also come with limitations. In particular, there is inherent difficulty to represent dissipative processes, and the restrictions placed on end point variations are not consistent with the definition of initial value problems. The present work on the time-domain response of poroelastic media extends the recent formulations of the mixed convolved action (MCA). The action in this proposed approach is formed by replacing the inner product in Hamilton's principle with a time convolution. As a result, dissipative processes can be represented in a natural way and the required constraints on the variations are consistent with the actual initial and boundary conditions of the problem. The variational formulation developed here employs temporal impulses of velocity, effective stress, pore pressure, and pore fluid mass flux as primary variables in this mixed approach, which also uses convolution operators and fractional calculus to achieve the desired characteristics. The resulting MCA is formulated directly in the time domain to develop a new stationary principle for poroelasticity, which applies to dynamic poroelastic and quasi-static consolidation problems alike. By discretizing the MCA using the finite element method over both space and time, new computational mechanics formulations are developed. Here, this formulation is implemented for the two-dimensional case, and several numerical examples of dynamic poroelasticity are presented to validate the approach.